The tight approximation property

This article introduces and studies the tight approximation property, a property of algebraic varieties defined over function field complex or real curve that refines weak (and known cohomological obstructions to it) by incorporating an condition in Euclidean topology. We prove is stable birational invariant, compatible with fibrations, satisfies descent under torsors linear groups. Its validity for number rationally connected follows. Some concrete consequences are: smooth loops locus compactification group, cubic hypersurface dimension at least 2, can be approximated rational curves; homogeneous spaces groups satisfy approximation.